Question

If $f(x)=\log _{a} x,$ show that $f\left(x^{\alpha}\right)=\alpha f(x)$.

   If $f(x)=\log _{a} x,$ show that $f\left(x^{\alpha}\right)=\alpha f(x)$.

College Algebra

College Algebra

Michael Sullivan 10th Edition

Chapter 6, Problem 108

Instant Answer

Solved by Expert Heather Zimmers

05/08/2020

Step 1

Step 1: First, we substitute $x^{\alpha}$ into the function $f(x)$, which gives us $f\left(x^{\alpha}\right)=\log _{a} x^{\alpha}$. Show more…

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If $f(x)=\log _{a} x,$ show that $f\left(x^{\alpha}\right)=\alpha f(x)$.

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Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, defined so that log base a of x gives the exponent to which a must be raised to obtain x. This concept is central in understanding the relationship between exponents and logarithms, and it provides a way to solve equations where the unknown is an exponent.

Power Rule of Logarithms

The power rule of logarithms states that the logarithm of a number raised to a power equals the power multiplied by the logarithm of the number, i.e., log_a(x^?) = ? · log_a(x). This rule simplifies expressions involving exponents inside logarithms and is derived from the properties of exponentiation.

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